5.2 CONCLUSIONS INVOLVING V AND +. BUT NOT 3 169
(b) Let C, be the curve in the xy plane described parametrically by x = cosh t
and y = sinh t. Use well-known properties of sinh and cosh to prove that C, =
{(cosh t, sinh t) I t E R} is a proper subset of ((x, y) E R x R 1x2 - y2 = 1) = C,.
(Recall the paragraph immediately following Example 11.)- A curve C described by an equation F(r, 8) = 0 in polar coordinates has the
property that a point with polar representation (r, 8) is on C if and only if an
ordered pair either of the form (r, 0 + 2nn) or of the form (- r, 0 + (2n + 1)n) satis-
fies the defining equation, for some positive integer n. (For most curves, it suffices
to consider the case n = 0.) Let curves C, and C2 in the xy plane be defined by
C, = {(r, 8)lr = cos 0 + 1) and C, = {(r, 8)lr = cos 8 - 1). Prove that C, = C,. - A subset S of the real line R is said to be convex if and only if, for all x, y E S
and for every real number t satisfying 0 I t 5 1, the real number tx + (1 - t)y is
an element of S.
*(a) Prove that [0, 11 is convex.
(b) Prove that [O,l] u [2,3] is not convex.
(c) Prove that if S, and S, are convex, then S, n S, is convex.
(d) Prove that if I is an interval in R, then I is convex. [In fact, the converse
of (d) is true as well. Its proof will be considered in Article 6.1.1
- (a) Suppose that T is a linearly independent subset of a vector space V and
that S E T. Prove that S is linearly independent.
(b) Let v,, v,, and V, be linearly independent vectors in a vector space V and let
c be a nonzero scalar. Prove that the sets {v,, cv,, v,) and {v, + cv,, v,, v,) are
also linearly independent. (Note: some familiarity with elementary properties of
vector addition and scalar multiplication is needed for this proof.) - A square matrix A = (aij), , is said to be a diagonal matrix if and only if, for
all i, j = 1,2,... , m, i # j implies aij = 0. A is upper (respectively, lower) triangular
if and only if i > j (respectively, i < j) implies aij = 0 for all i, j = 1,2,... , m.
(a) Give examples of a:
(i)^3 x 3 diagonal matrix
(ii)^4 x^4 upper triangular matrix
(iii) 3 x 3 matrix that is lower triangular and not diagonal.
(b) Prove that an m x m diagonal matrix is necessarily upper triangular.
(c) Prove that an m x m diagonal matrix is necessarily lower triangular. - Critique and complete (recall instructions in Exercise 1 1, Article 4.1).
(a) FACT The subset C = {(x, xS + x3) Ix E R) is symmetric with respect to the
origin.
"Proof" Let x = 3 and note that the point (3,270) E C, since y = 270 = 3= + 3,.
Note also that (-3, -270) E C, since -270 = -243 - 27 = (-3)5 + (-3),.
Since (x, y) and (-x, -y) are both on C, the desired symmetry is established.
(b) THEOREM A linear function y = f(x) = Mx + B is increasing on R if and
only if M > 0.
"Proof" We may prove the desired equivalence by proving implication in each
direction. In other words, we may prove that if M > 0, then f is increasing on R
if f is increasing on R, then M > 0. (=>) Done in Example 7. (e) Let x,